The p-Value in One-Sample Tests for the Mean

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In reporting the result of a significance test, showing the observed test statistic and its location in the null distribution along with the tail area is a helpful and frequently used graphic. This Demonstration makes it easy to construct this plot for the one-sample test for the population mean of a normal distribution. The effect of sample size, the assumption that the population variance is known or not, as well the question of using a one-sided or two-sided test may also be explored.


Let be the sample mean of observations in a random sample from a normal population with mean and standard deviation .

The -value for testing against one of the alternatives , or is illustrated using tests based on the test statistic when is assumed known, and when is not known, where , and is the sample standard deviation.

The -value for the test is shown by the shaded area. When the tail area is smaller than 5%, a red arrow is used to indicate the position of the test statistic.

The null distribution for and are respectively the standard Gaussian and Student on degrees of freedom. So when is known, the slider for is disabled.


Contributed by: Ian McLeod (March 2011)
Open content licensed under CC BY-NC-SA



Snapshot 1: versus , assumed known and based on a sample of 10 observations, so

Snapshot 2: as in Snapshot 1, but unknown and , yielding

Snapshot 3: and , and , yielding . A two-sided test is used.

Snapshot 4: As in Snapshot 3, but suppose is based on

Snapshot 5: with known variance

The two-sided test is always more conservative and for this reason many researchers discourage the use of one-sided tests [1].

[1] G. van Belle, Statistical Rules of Thumb, 2nd ed., New York: Wiley, 2008 p. 16.

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